Probit model

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In statistics , a branch of mathematics , the probit model is the specification of a generalized linear model . Probit is a suitcase word for prob (ability un) it , which arose from the two overlapping English words for probability and unit (0 or 1).

The statistical specification describes the process of model development in which a statistically estimable model (estimation model) is established. Generalized linear models are nonlinear extensions of classical linear regression . The probit model uses a probit coupling function that relates the expected value of the target variable to the linear predictor (prior knowledge) of the model. Chester Bliss introduced Probit models .

application

Like the logit models , the probit models are used to map binary target variables in binary discrete decision models . You use target values ​​that can only have two values. Examples: ${\displaystyle Y}$

"Getting divorced" → Yes / No,

"Customer has bought product A" → Yes / No, ${\displaystyle i}$

${\displaystyle X}$ → . ${\displaystyle Y}$

As a sample, customers are asked at the exit whether they have bought product A. The probit model can first - analogous to the regression - calculate whether the simultaneously collected characteristics explain the purchasing behavior “well”. In the positive case, it is possible to estimate how big the sales are when describing the entire market. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X}$

These models are very widespread in use. Within the generalized linear models, the logit model gives better results at extremely independent variable levels. Conversely, the probit model is generally better for random effects with medium-sized datasets.

definition

Probit models are econometric , non-linear models for explaining binary target variables with the coding: 0 = event does not occur, 1 = event occurs. The vector of the explanatory variables stands for the various observations that can be differentiated using the index . It influences the probability of whether event 0 or 1 will occur. Be the target and the influencing factor. ${\displaystyle \mathbf {x} _{i}}$ ${\displaystyle i}$ ${\displaystyle Y}$ ${\displaystyle X}$

The probit model is a clever definition in model development and is a formula:

${\displaystyle \Pr(Y=1|X=x)=\Phi (\mathbf {x} _{i}'{\boldsymbol {\beta }})\;{\stackrel {\mathrm {def} }{=}}\;{\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\mathbf {x} _{i}'{\boldsymbol {\beta }}}\operatorname {exp} \left(-{\frac {1}{2}}t^{2}\right)\mathrm {d} \,t}$ ,

Notation:

• ${\displaystyle \Phi (x)}$, pronounced “Phi of x”, describes the distribution function of a standard normal distribution with the probability that the associated random variable assumes a value less than or equal to. ${\displaystyle X}$ ${\displaystyle x}$
• The normalizing constant belongs to the integral from minus infinity to , written using the exponential function , and is a bound variable . ${\displaystyle {\frac {1}{\sqrt {2\pi }}}}$ ${\displaystyle x}$ ${\displaystyle \int _{-\infty }^{x}\mathrm {d} \,t}$ ${\displaystyle \operatorname {exp} ()}$ ${\displaystyle t}$
• The non-elementary integral is necessary to normalize the normal distribution density to the probability density . It was developed by Pierre-Simon Laplace in 1782 .

The formula for the probit model is: The probability “ ” - depending on the explanatory variables - that the response variable is the same corresponds to a function with the linear combination of the explanatory variables . The parameter vector is typically estimated using the maximum likelihood method . With this method of greatest density, that vector is selected as the estimate, according to whose distribution the realization of the observed data appears most plausible. ${\displaystyle \Pr }$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle 1}$ ${\displaystyle \Phi (x)}$ ${\displaystyle \mathbf {x} _{i}'{\boldsymbol {\beta }}}$ ${\displaystyle {\boldsymbol {\beta }}}$ ${\displaystyle {\boldsymbol {\beta }}}$ ${\displaystyle Y}$

model

The probit model is a simple latent variable model that describes the relationship between observable (or manifest) variables and the underlying latent variables . The term can have small errors . That is why it is replaced by : ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle \mathbf {x} _{i}'{\boldsymbol {\beta }}}$ ${\displaystyle \varepsilon _{i}}$ ${\displaystyle y_{i}^{*}}$

${\displaystyle y_{i}^{*}\;{\stackrel {\mathrm {def} }{=}}\;\mathbf {x} _{i}'{\boldsymbol {\beta }}+\varepsilon _{i}}$,

where the error terms follow a normal distribution with . They are similar to the well-known Gaussian distribution with the mean and the standard deviation . In addition, there is a dummy variable (yes-no variable), which is an indicator of whether the latent variable is positive: ${\displaystyle \varepsilon _{i}}$ ${\displaystyle \varepsilon _{i}\sim {\mathcal {N}}(0,1)}$ ${\displaystyle {\mathcal {N}}}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle Y}$ ${\displaystyle y_{i}^{*}}$

${\displaystyle Y\;{\stackrel {\mathrm {def} }{=}}\;1\Leftrightarrow y^{*}>0}$.

Then one can show that the following equation for the probit model is fulfilled:

${\displaystyle \Pr(Y=1|X=x)=\Phi (\mathbf {x} _{i}'{\boldsymbol {\beta }})}$.

Individual evidence

1. ↑ Oxford English Dictionary , 3rd ed. Sv probit (article dated June 2007): CI Bliss: The Method of Probits . In: Science . 79, No. 2037, 1934, pp. 38-39. doi : 10.1126 / science.79.2037.38 . PMID 17813446 . "These arbitrary probability units have been called 'probits'."